Change of Rate

Question

A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.4 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 16 cm. (Note the answer is a positive number).

Answer 1

Let V = volume at time t

      D = diameter at time t

      r = D/2 = radius at time t

 

Given:  dD/dt = -0.4

Find:    dV/dt when D = 16

 

V = (4/3)πr³ = (4/3)π(D/2)³

 

                      = (1/6)πD³

 

     dV/dt   = (1/2)πD²(dD/dt)

 

                  = (1/2)π(16)²(-0.4)

 

                  ≈ -160.8 cm³/min

 

The negative sign in front of 160.8 indicates that the volume is decreasing at the rate of

160.8 cm³/min